The infinite solenoid has a uniform magnetic field parallel to the solenoid axis. Let that axis be the z axis, for concreteness. So [itex]\mathbf {B}=B \mathbf {k}[/itex]. Now the vector potential is [itex]\mathbf {B}=\nabla \times \mathbf {A}[/itex]. The vector potential is not unique, so any vector-valued function [itex]\mathbf {A}[/itex] whose curl gives you [itex]B \mathbf{k}[/itex] will fit the bill.

How to find the Vector Potential of an infinite solenoid with n turns per unit length,radius R and current I.
since here current extends to infinty..
How will it be done
Pls Help

The line integral of A around a closed path is equal to the flux of B through the path. For an infinitely long solenoid, B = (n)(I)/((eps0)(c^2)) at internal points. (B = 0 at all points outside of the solenoid.) Thus the flux of B inside the solenoid is (pi)(R^2)(n)(I)/((eps0)(c^2)). At a distance r>R from the solenoid's axis, (2)(pi)(r)(A)=flux of B. That is,